Problem $4$

Consider a long conveyor belt $($labeled as a wall in the figure$),$ moving in the $+y$ direction at constant speed V $,$ from the fluid bath. The thickness $h$ of the film is constant within the region of interest, and the flow within this region is fully developed. You may assume the flow is laminar. If you use the incompressible Navier Stokes or mass continuity equations as a starting point, you must justify terms that you cancel or can neglect.

a$)$ Prove that the $x-$velocity component is zero within the region of interest.

b$)$ Simplify the veitical momentum equation $($below$)$ to obtain an ordinary differential equation for the velocity field. In order to receive credit, you must justify your simplifications. Also include the boundary conditions needed to solve the ODE.

$(\frac{del{u}_y}{\text{d}\text{e}\text{l}\text{t}}+u_x\frac{del{u}_y}{\text{d}\text{e}\text{l}\text{x}}+u_y\frac{del{u}_y}{\text{d}\text{e}\text{l}\text{y}}+u_z\frac{del{u}_y}{\text{d}\text{e}\text{l}\text{z}})=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{y}}+(\frac{de{l}^2{u}_y}{del{x}^2}+\frac{de{l}^2{u}_y}{del{y}^2}+\frac{de{l}^2{u}_y}{del{z}^2})+g_y$

c$)$ Solve for the vertical velocity field in the fully developed region. Show detailed work.